Integrals

1. If \(f(x,y)\) can be written as the product of two functions that depend only on \(x\) and \(y\), that is \(f(x,y)=g(x)h(y)\). Then

$$\int^b_a\int^d_cf(x,y)dydx=\int^b_ag(x)dx\int^d_ch(y)dy$$

2. Polar coordinate

$$\iint_Df(r,\theta)dA=\iint_Df(r,\theta)rdrd\theta$$

3. Cylindrical coordinate

$$\iiint_Vf(r,\theta,z)dV=\iiint_Vf(r,\theta,z)rdrd\theta dz$$

$$\begin{cases}
r^2=x^2+y^2\\
x=r\cos \theta\\
y=r\sin \theta\\
z=z
\end{cases}$$

4. Spherical Coordinate

$$\iiint_Vf(\rho,\theta,\phi)dV=\iiint_Vf(\rho,\theta,\phi)\rho^2\sin \phi d\rho d\theta d\phi$$

$$\begin{cases}
r^2=x^2+y^2+z^2\\
x=\rho\sin\phi\cos \theta\\
y=\rho\sin\phi\sin \theta\\
z=\rho\cos\phi
\end{cases}$$

5. Changing coordinates

$$dxdy=|\det\begin{pmatrix}
\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\
\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}
\end{pmatrix}
|dudv$$

The matrix is called the Jacobian matrix.

6. Line integral. If \(C:x(t)\vec i+y(t)\vec j, t\in[a,b]\)

$$\int_C f(x,y)ds=\int^b_af(x(t),y(t))\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$$

$$\int_C f(x,y)dx=\int^b_af(x(t),y(t))x'(t)dt$$

7. Vector field line integral. If \(\vec F=(P,Q,R)\),

$$\int_C \vec F\cdot d\vec{r}=\int_C Pdx+Qdy+rdz=\int_C \vec F(\vec r(t))\cdot \vec{r}'(t)dt$$

8. If the function is the gradient of another function

$$\int_C \nabla f\cdot d\vec{r}=f(end)-f(start)$$

9. A vector field is conservative if the line integral is independent of paths, or the function is the gradient of another function, or \(\oint\vec F\cdot d\vec r=0\)

10. In a simple connected region,

$$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x} \iff \vec F \ is \ conservative$$

11. Green’s theorem. For a simple connected region, the counterclockwise line integral around the edge of the region can be written as a double integral over that region.

$$\oint_C \vec F\cdot d\vec r=\oint_C Pdx+Qdy=\iint_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$$

The area of a region can be calculated by

$$A=\oint_Cxdy=-\oint_Cydx=\frac 1 2\oint_Cxdy-ydx$$

12. The surface area formula for functions with the form \(f(x,y)\)

$$S=\iint_D\sqrt{1+(\frac{\partial f}{\partial x})^2+(\frac{\partial f}{\partial y})^2} dA$$

13. The surface equation in parametric form

$$\vec r(u,v)=x(u,v)\vec i+y(u,v)\vec j+z(u,v)\vec k$$

14. The normal vector of a parametric surface, let

$$\vec r_u=\frac{\partial x}{\partial u}\vec i+\frac{\partial y}{\partial u}\vec j+\frac{\partial z}{\partial u}\vec k$$

$$\vec r_v=\frac{\partial x}{\partial v}\vec i+\frac{\partial y}{\partial v}\vec j+\frac{\partial z}{\partial v}\vec k$$

$$\vec n=\vec r_u\times \vec r_v$$

The unit normal vector is

$$\vec n=\frac{\vec r_u\times \vec r_v}{|\vec r_u\times \vec r_v|}$$

If the surface has the form \(\vec r(u,v)=x\vec i+y\vec j+z(x,y)\vec k\)

$$\vec n=\frac{-\frac{\partial z}{\partial x}\vec i-\frac{\partial z}{\partial y}\vec j+\vec k}{\sqrt{(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2+1}}$$

The unit normal vectors always points upwards since the z-component is always positive.

For a close surface, the unit normal vector is defined to point outwards

16. The surface area of a parametric surface

$$S=\int^b_a\int^d_c |\vec r_u \times \vec r_v|dvdu$$

17. Surface Integral$$\iint_S fdS=\iint_D f(\vec r(u,v))|\vec r_u\times\vec r_v|dA$$$$=\iint_D f(x,y,z)\sqrt{(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2+1}dA$$

18. Flux

$$\iint_S \vec F\cdot d\vec S=\iint_S \vec F\cdot \vec ndS=\iint_D \vec F(\vec r(u,v))\cdot(\vec r_u\times\vec r_v)dA$$$$=\iint_D (-P\frac{\partial z}{\partial x}-Q\frac{\partial z}{\partial y}+R)dA$$

19. If the surface is a sphere, that is \( \vec r(\theta, \phi) = (R\sin\phi\cos\theta,R\sin\phi\sin\theta,R\cos\phi)\)

$$\vec r_{\theta}\times \vec r_{\phi}=-R^2\sin\phi(\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi) $$

$$|\vec r_{\theta}\times \vec r_{\phi}|=R^2\sin\phi$$